Size-dependent buckling analysis of non-prismatic Timoshenko nanobeams made of FGMs rested on Winkler foundation

Document Type : Research

Authors

1 Assistant Professor, Department of civil engineering, University of Kashan, Kashan, Iran.

2 MSc Student in Structural Engineering, Department of Civil Engineering, Faculty of Engineering, Shahr-e-Qods Branch, Islamic Azad University.

Abstract

In this article, the buckling behavior of tapered Timoshenko nanobeams made of axially functionally graded (AFG) materials resting on Winkler type elastic foundation is perused. It is supposed that material properties of the AFG nanobeam vary continuously along the beam’s length according to the power-law distribution. The nonlocal elasticity theory of Eringen is employed to contemplate the small size effects. Based on the first-order shear deformation theory, the system of nonlocal equilibrium equations in terms of vertical and rotation displacements are derived using the principle of total potential energy. To acquire the nonlocal buckling loads, the differential quadrature method is used in the solution of the resulting coupled differential equations. Eventually, an exhaustive numerical example is carried out for simply supported end conditions to investigate the influences of significant parameters such as power-law index, tapering ratio, Winkler parameter, aspect ratio, and nonlocal parameter on the buckling capacity of AFG Timoshenko nanobeams with varying cross-section supported by uniform elastic foundation.

Keywords

References

1. Yang FA, Chong AC, Lam DC, Tong P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures. 2002; 39(10):2731-43. [DOI:10.1016/S0020-7683(02)00152-X]
2. Gurtin ME, Weissmüller J, Larche F. A general theory of curved deformable interfaces in solids at equilibrium. Philosophical Magazine A. 1998; 78(5):1093-109. [DOI:10.1080/01418619808239977]
3. Eringen AC. Nonlocal polar elastic continua. International journal of engineering science. 1972; 10(1):1-6. [DOI:10.1016/0020-7225(72)90070-5]
4. Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 1983; 54: 4703-4710. [DOI:10.1063/1.332803]
5. Elishakoff I, Guede Z. Analytical polynomial solutions for vibrating axially graded beams. Mechanics of Advanced Materials and Structures. 2004; 11(6):517-33. [DOI:10.1080/15376490490452669]
6. Reddy JN. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science. 2007; 45(2-8):288-307. [DOI:10.1016/j.ijengsci.2007.04.004]
7. Wang CM, Zhang YY, He XQ. Vibration of Non-local Timoshenko Beams. Nanotechnology 2007; 18: 1-9. [DOI:10.1088/0957-4484/18/10/105401]
8. Aydogdu M. A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E: Low-dimensional Systems and Nanostructures. 2009; 41(9):1651-5. [DOI:10.1016/j.physe.2009.05.014]
9. Civalek Ö., Akgöz B. Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler-Bernoulli beam modeling. Scientia Iranica-Transaction B: Mechanical Engineering, 2010; 17(5): 367-375.
10. Danesh, M., Farajpour, A., Mohammadi, M., Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method. Mechanics Research Communications, 2012; 39(1): 23-27. [DOI:10.1016/j.mechrescom.2011.09.004]
11. Wang, B.L., Hoffman, M. and Yu, A.B., 2012. Buckling analysis of embedded nanotubes using gradient continuum theory. Mechanics of Materials, 45, pp.52-60. [DOI:10.1016/j.mechmat.2011.10.003]
12. Eltaher MA, Alshorbagy AE, Mahmoud FF. Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Composite Structures. 2013; 99:193-201. [DOI:10.1016/j.compstruct.2012.11.039]
13. Eltaher MA, Emam SA, Mahmoud FF. Static and stability analysis of nonlocal functionally graded nanobeams. Composite Structures. 2013; 96:82-8. [DOI:10.1016/j.compstruct.2012.09.030]
14. Akgöz B, Civalek Ö. Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory. Composite Structures, 2013; 98: 314-322. [DOI:10.1016/j.compstruct.2012.11.020]
15. Ke LL, Wang YS. Free vibration of size-dependent magneto-electro-elastic nanobeams based on the nonlocal theory. Physica E: Low-Dimensional Systems and Nanostructures. 2014; 63:52-61. [DOI:10.1016/j.physe.2014.05.002]
16. Pandeya A, Singhb J. A variational principle approach for vibration of non-uniform nanocantilever using nonlocal elasticity theory. Proced. Mater. Sci. 2015; 10: 497-506. [DOI:10.1016/j.mspro.2015.06.087]
17. Shafiei N, Kazemi M, Safi M, Ghadiri M. Nonlinear vibration of axially functionally graded non-uniform nanobeams. International Journal of Engineering Science, 2016; 106: 77-94. [DOI:10.1016/j.ijengsci.2016.05.009]
18. Akgöz, B. and Civalek, Ö. 2016. Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronautica, 119, pp.1-12. [DOI:10.1016/j.actaastro.2015.10.021]
19. Ghasemi, A.R. and Mohandes, M., 2016. The effect of finite strain on the nonlinear free vibration of a unidirectional composite Timoshenko beam using GDQM. Advances in aircraft and spacecraft science, 3(4), p.379. [DOI:10.12989/aas.2016.3.4.379]
20. Mohandes, M. and Ghasemi, A.R., 2016. Finite strain analysis of nonlinear vibrations of symmetric laminated composite Timoshenko beams using generalized differential quadrature method. Journal of Vibration and Control, 22(4), pp.940-954. [DOI:10.1177/1077546314538301]
21. Ebrahimi F, Salari E. Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment, Acta Astronautica, 2015; 113: 29-50. [DOI:10.1016/j.actaastro.2015.03.031]
22. Ebrahimi F, Barati M R. Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium, J. Brazil. Soc. Mech. Sci. Eng. 2016; 39(3); 937-952. [DOI:10.1007/s40430-016-0551-5]
23. Ebrahimi F, Barati M R. A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams, Compos. Struct. 2017; 159: 174-182. [DOI:10.1016/j.compstruct.2016.09.058]
24. Mercan, K. and Civalek, Ö, 2016. DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Composite Structures, 143, pp.300-309. [DOI:10.1016/j.compstruct.2016.02.040]
25. Calim, F.F., 2016. Free and forced vibration analysis of axially functionally graded Timoshenko beams on two-parameter viscoelastic foundation. Composites Part B: Engineering, 103, pp.98-112. [DOI:10.1016/j.compositesb.2016.08.008]
26. Lezgy-Nazargah, M., Vidal, P. and Polit, O., 2013. An efficient finite element model for static and dynamic analyses of functionally graded piezoelectric beams. Composite Structures, 104, pp.71-84. [DOI:10.1016/j.compstruct.2013.04.010]
27. Lezgy-Nazargah, M. and Farahbakhsh, M., 2013. Optimum material gradient composition for the functionally graded piezoelectric beams. International Journal of Engineering, Science and Technology, 5(4), pp.80-99. [DOI:10.4314/ijest.v5i4.8]
28. Lezgy-Nazargah, M., 2015. A three-dimensional exact state-space solution for cylindrical bending of continuously non-homogenous piezoelectric laminated plates with arbitrary gradient composition. Archives of Mechanics, 67(1), pp.25-51.
29. Lezgy-Nazargah, M., 2016. A three-dimensional Peano series solution for the vibration of functionally graded piezoelectric laminates in cylindrical bending. Scientia Iranica. Transaction A, Civil Engineering, 23(3), p.788. [DOI:10.24200/sci.2016.2159]
30. Lezgy-Nazargah, M., 2015. Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach. Aerospace Science and Technology, 45, pp.154-164. [DOI:10.1016/j.ast.2015.05.006]
31. Lezgy-Nazargah, M. and Meshkani, Z., 2018. An efficient partial mixed finite element model for static and free vibration analyses of FGM plates rested on two-parameter elastic foundations. Struct Eng Mech, 66, pp.665-676.
32. Demir C, Mercan K, Numanoglu H M, Civalek Ö. Bending response of nanobeams resting on elastic foundation. J. Appl. Comput. Mech. 2018; 4(2): 105-11.
33. Soltani, M. and Mohammadi, M., 2018. Stability Analysis of Non-Local Euler-Bernoulli Beam with Exponentially Varying Cross-Section Resting on Winkler-Pasternak Foundation. Journal of Numerical Methods in Civil Engineering, 2(3), pp.67-77.
34. Ghannadiasl, A., 2019. Natural frequencies of the elastically end restrained non-uniform Timoshenko beam using the power series method. Mechanics Based Design of Structures and Machines, 47(2), pp.201-214. [DOI:10.1080/15397734.2018.1526691]
35. Soltani, M. and Asgarian, B., 2019. New hybrid approach for free vibration and stability analyses of axially functionally graded Euler-Bernoulli beams with variable cross-section resting on uniform Winkler-Pasternak foundation. Latin American Journal of Solids and Structures, 16(3). [DOI:10.1590/1679-78254665]
36. Ghasemi, A.R. and Mohandes, M., 2019. A new approach for determination of interlaminar normal/shear stresses in micro and nano laminated composite beams. Advances in Structural Engineering, 22(10), pp.2334-2344. [DOI:10.1177/1369433219839294]
37. Arefi, M. and Civalek, O., 2020. Static analysis of functionally graded composite shells on elastic foundations with nonlocal elasticity theory. Archives of Civil and Mechanical Engineering, 20(1), pp.1-17. [DOI:10.1007/s43452-020-00032-2]
38. Soltani, M. and Asgarian, B., 2020. Lateral-Torsional Stability Analysis of a Simply Supported Axially Functionally Graded Beam with a Tapered I-Section. Mechanics of Composite Materials, pp.1-16. [DOI:10.1007/s11029-020-09859-5]
39. Bellman R. E, Casti J., "Differential quadrature and long-term integration." Journal of Mathematical Analysis and Applications; 34, 235-238 (1971). [DOI:10.1016/0022-247X(71)90110-7]
40. Soltani M. Finite element modelling for buckling analysis of tapered axially functionally graded Timoshenko beam on elastic foundation. Mechanics of Advanced Composite Structures, DOI: 10.22075/MACS.2020.18591.1223.
Numerical Methods in Civil Engineering
Volume 3, Issue 2
December 2018
Pages 35-46

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History

  • Receive Date: 03 August 2018
  • Revise Date: 05 October 2018
  • Accept Date: 10 November 2018

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